Our paper describes intertemporal interactions between vampires and humans based on several types of vampire behavior described in popular fiction, films, and television series. Our main research question is: If vampires were real and lived among us, would their existence be possible? We draw several scenarios of vampire–human equilibria and use models with differential equations to test under what provisions vampires could have existed amongst humans. Mathematical modeling reveals that several popular culture sources outline the models describing plausible and peaceful coexistence.

Recent Research About Vampires

Since the 1980s, such topics as behavior of vampires, economic significance of vampirism, and optimal bloodsucking strategies (e.g. preventing the depletion of renewable human resources) have found their way into the research literature, becoming an inspiration for several academic papers (Hart and Mehlmann, 1982, 1983; Hartl, Mehlmann and Novak, 1992; Neocleus, 2003; Efthimiou and Gandhi, 2007).

Vampires are often described in legends and folklore. The word "vampire" comes from the Hungarian language. The first myths and legends about vampires can be found in Mesopotamian texts dating back to 4000 B.C.E. (Campbell Thompson, 1904).

Consider introducing vampires into the model of population growth denoted by dx/dt = kx. The vampire population is denoted by the function y(t), y0=1. Vampires act as natural predators for humans. The human population dynamics can therefore be presented as the following function: dx/dt = kx-v(x)y, where v(x) is the rate at which humans are killed by vampires.

Assume that the number of any vampire's victims is growing proportionally. Thence, the function v(x) can be presented as the following: v(x)=ax, where a > 0 is the coefficient of the human's lethal interaction with a vampire (a human is either killed by a vampire or is turned into a vampire). As a result, the differential equation describing the growth rate of human population can be formulated as the following: dx/dt = x(k-ay). Assume the dynamics of vampire's population change to be y(t). The growth of vampire population will be determined by the quality and quantity of interactions with humans.

After selecting its victim, any vampire can kill it by draining its blood, turn it into a new vampire, or feed on it but leave it to live.

Let us also introduce vampire slayers into the model. The slayers regulate the population of vampires by periodically killing vampires. The equation will then be modified to be dy/dt = baxy-cy, where 0 < b ≤ 1 is the coefficient reflecting the rate with which humans are turned into vampires and c ≥ 0 is the coefficient of lethal outcome of the interaction between a vampire and vampire slayer.

In order to solve this, we need to consider a Lotka-Volterra system, or a "predator–prey" type model (Volterra, 1931). The system allows for the stationary solution, meaning that there is a pair of solutions for the system that creates a state when human and vampire populations can coexist in time without any change in numbers. The size of human population is determined by the effectiveness of slaying vampires by vampire hunters c and the number of cases when the humans are turned into vampires ba. The size of vampire population depends on the growth rate of human population k and vampires' thirst for human blood a. The stationary solution shows that when vampires are capable of restraining their blood thirst, the size of both populations can be rather high in mutual co-existence. The system is held in balance by the existence of vampire slayers.

The Stoker-King model

Bram Stoker's Dracula and Stephen King's Salem's Lot describe interactions between vampires and humans in the following way: A vampire selects a human victim and gets into the victim's proximity. This typically happens after dark. Sometimes the vampire needs the victim to invite the vampire in, but often the vampire does not require permission to enter the victim's premises and attacks the sleeping victim (Stoker, 1897; McNally and Florescu, 1994). The vampire bites the victim and drinks the victim's blood and returns to feed for 4–5 consecutive days, whereupon the victim dies, is buried, and rises to become another vampire (unless a wooden stake is put through the new vampire's heart). Vampires usually need to feed every day, so more and more human beings are constantly turned into vampires (Stoker, 1897; King, 1975).

Assume the events described in Dracula were real. How would things evolve given the Stoker-King model dynamics described in both sources? Let us take 1897 as the starting point (the year Stoker's novel was published). In 1897, the world population was about 1.65 billion people (UN, 1999). The model is presented on Diagram 1.

Let us calibrate the parameters of this specific case of predator–prey model. The calculation period is set at 1 year with a step of 5 days (t = 0 ... 73). The coefficient of human population growth k for the given period is very small and can be neglected, therefore k = 0. The coefficient of lethal outcome for humans interacting with vampires can be calculated according to the scenario presented in the Stoker-King model y0(t) = y0qt, where y0 = 1, q = 2. The probability of a human (who interacts with a vampire) being turned into a vampire is very high, thence b = 1. Jonathan Harker and Abraham van Helsing could not be considered very efficient vampire slayers; therefore we can put c = 0.

The resulting simplified model is presented in a form of a Cauchy problem. Due to the fact that the total sum of humans and vampires does not change in time (human population does not grow and humans gradually become vampires), the predator-prey model is diminished to a simple problem of an epidemic outbreak (Munz et al., 2009).

The solution to this problem is presented in Chart 1. It is clearly visible that the human population is drastically reduced by 80% by the 165th day from the moment when the first vampire arrives. At the end of our one-year study window, the world will be inhabited by 1,384 million vampires and 266 million people.

It is obvious that the growth of vampire population is extreme: at first, the number of vampires jumps up abruptly, but then slows down and declines. The maximal growth of the number of vampires (infected humans) will be observed on the 153rd day, when the number of vampires is the highest and equals 825 million with 286 million newly turned vampires every day. It is apparent that the increase in one population (vampires) inevitably leads to the decrease in another (humans). The presence of vampires in the Stoker– King model brings the mankind to the brink of extinction and the model becomes very similar to an epidemic outbreak caused by a deadly virus (e.g. Ebola or SARS). According to the Stoker–King model, vampires need just half a year to take up man's place on Earth.

The Harris-Meyer-Kostova model

Stephenie Meyer's Twilight series of books, Charlaine Harris's Sookie Stackhouse (Southern Vampire) series of books (turned into the True Blood television series), and Elizabeth Kostova's novel The Historian show worlds where vampires peacefully coexist with humans.

In Meyer's Twilight series, vampires can tolerate the sunlight, interact with humans (even fall in love with them), and drink animal blood to survive (Meyer, 2005), but they have to live in secrecy and pretend to be human beings. In the Sookie Stackhouse books and True Blood television series, however, vampires and humans live side by side and are aware of each other. Vampires can buy synthetic human blood of different blood types that is sold in bottles and can be bought in every grocery store, bar, or gas station (Harris, 2001). They cannot walk during daytime, so they usually come out at night. Vampire blood is a powerful hallucinogenic drug that is sought by humans and traded on the black market. (Sometimes humans capture vampires with the help of silver chains or harnesses and then kill them by draining their blood.) Some humans seek sex with vampires, as vampires are stronger and faster than humans and can provide superb erotic experiences. There is a possibility to turn a human being into a vampire, but it takes time and effort.

In The Historian, vampires are rare and do not reveal themselves to humans too often. Their food ratios are limited and they spend lots of time brooding in their well-hidden tombs (Kostova, 2005).

In Harris's series, vampires have decided to reveal themselves to humans and coexist with them, peacefully exerting their citizens' rights (Harris, 2001). Assume that at the time of the events described in the first book of the series, Dead Until Dark (2001), the world's vampire hypothetical population was around 5 million, the population of the state of Louisiana in 2001 (Maddison, 2006). The initial conditions of the Harris-Meyer-Kostova model are therefore the following: 5 million vampires, 6159 million people, organized groups of vampire "drainers". The model is presented in Diagram 2.

Humans almost always come out alive from their encounters with vampires, hence the coefficient of lethal outcome a = 0.01. The probability of a human being turned into a vampire is b = 0.1. There are numerous groups of vampire drainers (although the number of drained vampires is relatively low and would not lead to their total extinction), so we can put c > 0 (c is calculated similarly to the coefficient k). The model allows for a stationary solution: there are system parameters (xs, ys) that would stabilize the populations of humans and vampires in time. In order to find the stabilized populations of both species, xs and ys, the equality is: (xs, ys) = (7704.8) million individuals. Chart 2 shows us the stationary solution presented on a logarithmic scale.

This stationary solution for 2001 cannot be found with the chosen population growth coefficient k and can be reached applying some conditions only after 2012. The deviations in the number of people and vampires from the stationary state at the initial period of time are quite small, which points at the fact that the system might be stable and auto-cyclical.

Our calculations yield that the human population will be growing until 2046 when it reaches its peak of 9.6 billion people, whereupon it will be declining until 2065 until it reaches its bottom at 6.12 billion people. This process will repeat itself continuously. The vampire population will be declining until 2023 when it reaches its minimum of 289 thousand vampires, whereupon it will be growing until 2055 until it reaches its peak at 397 million vampires. This process will repeat itself continuously. Chart 3 shows the phase diagram of the cyclical system of human–vampire co-existence. Under certain conditions, the Harris-Meyer- Kostova model seems plausible and allows for the existence of vampires in our world. Peaceful coexistence of two species is a reality.

The Whedon model

The television series Buffy the Vampire Slayer, created by Joss Whedon, presents the most simplistic, yet the most dreadful doomsday scenario of vampire–human interaction (similar to zombie infection outbreak in movies like 28 Days Later or Resident Evil and described in Munz et al. (2009)). The vampire bites its victim, who (in a very short period of time) rises as another undead vampire and, in turn, bites another human victim, and so on. Luckily enough for humans, the world is populated by an unknown (but considerably large) number of vampire slayers, with a girl named Buffy Summers being their most remarkable representative, and killing a vampire is relatively easy.

The Whedon model is a modified version of the Harris-Meyer-Kostova model. It uses the higher coefficient of vampire-slaying effectiveness, c. The initial conditions of the Whedon model are: 5 million vampires, 6159 million people, organized groups of vampire slayers. The model is presented in Diagram 3.

Let us calibrate the parameters of the model. The calculation period is set at 10 years with a step of 1 year, and the coefficient of human population's growth is calculated as k = ln(x1/x0)/t1-t0 where x1 = 7000 million people in 2012, x0 = 6150 million people at time t0 = 2001. Humans are always turned into new vampires after their encounters with vampires, so the coefficient of lethal outcome a is high. The probability of a human being turned into a vampire is b = 0.1. There are groups of vampire slayers, therefore we put c = 10. The resulting model is presented in Chart 4.

Although the Whedon model's structure theoretically allows for coexistence of humans and vampires, the laborious vampire slayers contribute to putting the system out of balance by killing all vampires. The human population recovers from the damage caused to it by vampires and continues to grow steadily.

Conclusions

Overall, it appears that although vampire–human interactions would in most cases lead to great imbalances in the ecosystems, there are several cases that might actually convey plausible models of coexistence between humans and vampires.

The Stoker-King model described the explosive rate of growth in vampire population that would lead to exterminating 80% of the human population on the 165th day of the first vampire's arrival. The scenario is similar to severe epidemic outbreaks and would lead first to the complete extinction of humans and then to the death of all vampires. The Harris-Meyer-Kostova model allows for the peaceful existence of vampires in our world. The Whedon model allows for the coexistence of humans and vampires, but in this case vampires become one of the endangered species due to the existence of super-effective vampire slayers. Unless the slayers calm their rigor, the vampire goes extinct.

A human stomach has, on average, a capacity of one liter. Most healthy humans can lose a liter of blood and survive, although being a bit woozy. For a vampire to drain a human completely, the vampire would have to fall under the model of a tick, that is mostly hollow or that can swell up with ingestion of fluid.

This always bugged me about the vampires on Buffy. Suck on a victim's neck for five seconds and they were dead.

Lotka-Volterra, is so far so good.. however, it should only work with species not capable of completely solving a problem...humans and smallpox have shown not to follow the Lotka-Volterra-equations as soon as one of them finally decided to bring it to an end. Eihter way, i rather believe that these species may only live separated by cages bars... the only question is which species would end in the smaller side of the cage. In the human role i would try to hunt all vampires down, (maybe keeping some vampire blood for scientific reasons)...In the vampire role i would try to breed a domesitcated homo sapiens, or other animal having a delicious type of blood...In both cases Lotka-Volterra won't be applicable.